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Geometry

Geometry. 13.1 The Distance Formula. A(-5, 4). . B(-1, 4). Example 1 Find the distance between the two points. a. A (–5, 4) and B (–1, 4) b. C (2, –5) and D (2, 7). | -5 – (-1) |. | 7 – (-5) |. 12 units. 4 units.

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Geometry

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  1. Geometry 13.1 The Distance Formula

  2. A(-5, 4). . B(-1, 4) Example 1 Find the distance between the two points. a. A(–5, 4) and B(–1, 4) b. C(2, –5) and D(2, 7) | -5 – (-1) | | 7 – (-5) | 12 units 4 units As you can see, if two points share an x coordinate or a y coordinate, the distance between the two points can be found by finding the distance (the absolute value of the difference) between the other coordinates. What if two points do not lie on a horizontal or vertical line?

  3. When two points do not lie on a horizontal or vertical line, you can find the distance between points by using the distance formula which is derived from the Pythagorean Theorem. The distance d between points and is: Why? Let’s try an example to find out! (-3, 4). Example 2 Find the distance between (–3, 4) and (1, –4). 4 . (1, -4) 4√5 8 Pythagorean Theorem!

  4. Use the distance formula to find the distance between the two points. 4. (4, 2) and (1,-1) Answer: 3√2

  5. Example 3 Given points A(1, 1), B(–3, –11), and C(4, 10), find AB, AC, and BC. Are A, B, and C collinear? If so, which point lies between the other two? AB = √(1+3)2 + (1+11)2 BC = √(-3 – 4)2 + (-11 - 10)2 AC = √(1 – 4)2 + (1 – 10)2 = √16+ 144 = √49+ 441 = √9+ 81 = √160 = √490 = √90 = 4√10 = 7√10 = 3√10 . . . Are A, B, and C collinear? If yes, the two small distances add to the large distance. 4√10 + 3√10 = 7√10 Since BC is the longest distance, B and C are the endpoints. The points are collinear!! Thus, point A must be in the middle.

  6. Pick one, #1 or #2!! 12. Show that the triangle with vertices P(5, 0), E(–1, –2), and T(3, 6) is isosceles. PE = √(5+1)2 + (0+2)2 ET = √(-1 – 3)2 + (-2 – 6)2 PT = √(5 – 3)2 + (0 – 6)2 = √36+ 4 = √16+ 64 = √4+ 36 = √40 = √80 = √40 = 2√10 = 4√5 = 2√10 The triangle is isosceles because two of its sides are the same length! 13. Quadrilateral GEMA has vertices G(3, 8), E(8, –3), M(–2, –5), and A(–5, 2). Show that its diagonals are congruent. GM = √(3+2)2 + (8+5)2 EA = √(8+5)2 + (-3 – 2)2 = √25+ 169 = √169+ 25 = √194 = √194 No need to simpilfy… The diagonals are congruent!

  7. An equation of the circle with center (a, b) and radius r is: This is the circle formula to remember!! Why is the circle formula in the section about the distance formula? If you rearrange the formula above it reads… Now that we know the distance formula, this represents all points that are “r distance” away from a point (a, b). This is precisely a circle!!!

  8. An equation of the circle with center (a, b) and radius r is: How could this be a circle? Let’s analyze (x – 0)2 + (y – 0)2 = 81 to see if it really is a circle!!

  9. Example 4 Find the center and radius of the circle with the equation: Center: (-3, 5) Radius = 2 Write an equation of the circle that has the given center and radius. 14. C(0, 0); r = 9 15. C(-3, -8); r = 6 16. C(1, -2); r = x2 + y2 = 81 (x + 3)2 + (y + 8)2 = 36 (x – 1)2 + (y + 2)2 = 3

  10. Find the center and radius of each circle. Sketch the graph. 17. 18. Center: (2, -4) Center: (-7, -3) Radius = 3 Radius = 5 . . 19. 20. . . Center: (1, 2) Center: (0, 1) Radius = 3/4 Radius = 7

  11. HW: P. 525-527 (CE 1-9 Odd; WE 1-29 Odd) Bring Compass • Good afternoon, my name is Sean Connors and I am with College Planning Specialists in Walnut Creek.  Below is a brief description of my company if you are not familiar with us: • We hold 2 to 3 Free-Community workshops every month at various Contra Costa County Library locations on how to Plan & Pay for college.  These workshops are 90 minutes in length and focus on the entire College Planning process from the students college/major/career choices to financially how to obtain maximum money for college.  Our goal is provide families with the framework of how to legally plan the students academic & parents financial situations to pay for college, all with as little out of pocket as possible.  We do not sell anything or offer our services at the workshop, it is strictly a platform to help families accomplish this on their own.  We have made available roughly 10-12 hours per week to meet with families that attend the workshop to meet with us to discuss their situations in further detail.  • Our intent is to help educate families about the current Academic & Financial Aid process, and how to help the parents & students plan for the future early in High School, typically Freshman or Sophomore year at the latest.  We understand the overwhelming and daunting task that Guidance Counselors have (especially in the MDUSD since they do not have them) with Counselor to Student ratios across the state of California at 1 Counselor to 966 students (CSAC 5/21/06) ranking us last (50th) in the Union.  We feel we can help alleviate the stress & pressure that many of these Counselors endure.  We have helped hundreds of families from an academic & financial standpoint.  We are also involved with various travel-sports teams & clubs in the Bay Area, providing College Placement for student athletes via our "Game-Plan" program. • Alexis and I have been conducting business for the past 3.5 years and I personally have been involved with the college search/planning/funding/scholarship process for the past 6 years.  I have been coaching football at Clayton Valley HS, Concord HS, and Diablo Valley College and have been fortunate to help many student-athletes obtain scholarships to continue their education at a 4-year institution and beyond.  • If you have any further questions or comments please feel free to contact me at anytime between 8-3pm, after 3pm I available only by email due to our student appointment hours of 3-7pm. • I look forward to your response and hope to meet with you soon! • Sean Connors, PresidentCollege Planning Specialists1901 Olympic Blvd. Suite 300Walnut Creek, CA 94596Direct ~ (925) 627-2648Email ~ sean@CollegeFundsNow.comWebsite ~ www.CollegeFundsNow.com

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